The ratio of radius of second orbit of hydrogen to the radius of its first orbit is :

Options

A. `2:1`
B. `1 : 2`
C. `4 : 1`
D. `6 : 1`

Answer

To find the ratio of the radius of the second orbit (R2) of hydrogen to the radius of its first orbit (R1), we can use Bohr's model of the atom. According to this model, the radius of the nth orbit (Rn) is given by the formula:

\[ R_n = \frac{0.529 \, n^2}{Z} \, \text{angstroms} \]

where:
- \( R_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (orbit number),
- \( Z \) is the atomic number (for hydrogen, Z = 1).

### Step-by-Step Solution:

1. **Identify the radii for the first and second orbits**:
- For the first orbit (n=1):
\[ R_1 = \frac{0.529 \, (1)^2}{1} = 0.529 \, \text{angstroms} \]
- For the second orbit (n=2):
\[ R_2 = \frac{0.529 \, (2)^2}{1} = \frac{0.529 \times 4}{1} = 2.116 \, \text{angstroms} \]

2. **Set up the ratio of the radii**:
\[ \frac{R_2}{R_1} = \frac{2.116}{0.529} \]

3. **Calculate the ratio**:
- Simplifying the ratio:
\[ \frac{R_2}{R_1} = \frac{2.116}{0.529} = 4 \]

4. **Express the ratio in simplest form**:
- Thus, the ratio of the radius of the second orbit to the radius of the first orbit is:
\[ R_2 : R_1 = 4 : 1 \]

### Final Answer:
The ratio of the radius of the second orbit of hydrogen to the radius of its first orbit is **4 : 1**.

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